U.S. patent application number 16/610070 was filed with the patent office on 2021-10-28 for 3d inversion of deep resistivity measurements with constrained nonlinear transformations.
The applicant listed for this patent is Halliburton Energy Services, Inc.. Invention is credited to Michael S. Bittar, Junsheng Hou.
Application Number | 20210333429 16/610070 |
Document ID | / |
Family ID | 1000005755741 |
Filed Date | 2021-10-28 |
United States Patent
Application |
20210333429 |
Kind Code |
A1 |
Hou; Junsheng ; et
al. |
October 28, 2021 |
3D INVERSION OF DEEP RESISTIVITY MEASUREMENTS WITH CONSTRAINED
NONLINEAR TRANSFORMATIONS
Abstract
A method includes setting a value of a formation parameter for a
subsurface formation and creating an initial three-dimensional (3D)
model of the subsurface formation based on the formation parameter.
The method also includes applying a constrained transformation to
one or more inversion variables of the initial 3D model to create a
variable-constrained 3D model of the subsurface formation and
applying an unconstrained minimization operation to the
variable-constrained 3D model to generate a first transformed 3D
model. The method also includes inverting the first transformed 3D
model to generate a first inverted 3D model of the subsurface
formation.
Inventors: |
Hou; Junsheng; (Kingwood,
TX) ; Bittar; Michael S.; (Houston, TX) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Halliburton Energy Services, Inc. |
Houston |
TX |
US |
|
|
Family ID: |
1000005755741 |
Appl. No.: |
16/610070 |
Filed: |
February 21, 2019 |
PCT Filed: |
February 21, 2019 |
PCT NO: |
PCT/US2019/018872 |
371 Date: |
October 31, 2019 |
Current U.S.
Class: |
1/1 |
Current CPC
Class: |
G06T 2219/2016 20130101;
G06T 17/20 20130101; G06T 19/20 20130101; G01V 3/38 20130101; G01V
99/005 20130101; E21B 2200/20 20200501; E21B 49/00 20130101; E21B
44/00 20130101 |
International
Class: |
G01V 3/38 20060101
G01V003/38; G01V 99/00 20060101 G01V099/00; E21B 44/00 20060101
E21B044/00; E21B 49/00 20060101 E21B049/00; G06T 17/20 20060101
G06T017/20; G06T 19/20 20060101 G06T019/20 |
Claims
1. A method comprising: setting a value of a formation parameter
for a subsurface formation; creating an initial three-dimensional
(3D) model of the subsurface formation based on the formation
parameter; applying a constrained transformation to one or more
inversion variables of the initial 3D model to create a
variable-constrained 3D model of the subsurface formation; applying
an unconstrained minimization operation to the variable-constrained
3D model to generate a first transformed 3D model; and inverting
the first transformed 3D model to generate a first inverted 3D
model of the subsurface formation.
2. The method of claim 1, wherein applying the constrained
transformation comprises applying a constrained nonlinear
transformation.
3. The method of claim 1, wherein the one or more inversion
variables comprise at least one of a horizontal resistivity, a
vertical resistivity, and an anisotropy of the subsurface
formation.
4. The method of claim 1, wherein the one or more inversion
variables comprise at least one of a formation dip and an azimuth
of the subsurface formation.
5. The method of claim 1, wherein the one or more inversion
variables comprise a boundary of a portion of the first inverted 3D
model.
6. The method of claim 1, further comprising drilling a borehole
into the subsurface formation based on the first inverted 3D
model.
7. The method of claim 1, wherein the initial 3D model is a first
initial 3D model, and wherein the method of claim 1 further
comprises: generating a first lower-dimensional model, wherein the
first lower-dimensional model is at least one of a one-dimensional
(1D) model and a two-dimensional (2D) model of the subsurface
formation based on the value of the formation parameter, wherein
creating the first initial 3D model comprises using the first
lower-dimensional model; generating a second lower-dimensional
model, wherein the second lower-dimensional model is different from
the first lower-dimensional model, and wherein the second
lower-dimensional model is as at least one of the 1D model and the
2D model of the subsurface formation based on the value of the
formation parameter; generating a second initial 3D model based on
the second lower-dimensional model; generating a second transformed
3D model based the second initial 3D model using the unconstrained
minimization operation; and inverting the second transformed 3D
model to generate a second inverted 3D model of the subsurface
formation.
8. The method of claim 1, further comprises generating a quality
indicator corresponding with a 3D portion of the first inverted 3D
model, wherein the quality indicator is based on at least one of a
standard deviation and relative error of the 3D portion.
9. The method of claim 1, wherein the unconstrained minimization
operation comprises using a Gauss-Newton algorithm.
10. The method of claim 1, wherein the unconstrained minimization
operation comprises using a Jacobian matrix, wherein the Jacobian
matrix is generated using at least one of a finite difference
method, an Adjoint method, a Broyden approximation, and a
combination thereof.
11. A system to generate a first inverted 3D model of a subsurface
formation, the system comprising: a processor; and a
machine-readable medium having program code executable by the
processor to cause the processor to, set a value of a formation
parameter for the subsurface formation; create an initial
three-dimensional (3D) model of the subsurface formation based on
the formation parameter; apply a constrained transformation to one
or more inversion variables of the initial 3D model to create a
variable-constrained 3D model of the subsurface formation; apply an
unconstrained minimization operation to the variable-constrained 3D
model to generate a first transformed 3D model; and invert the
first transformed 3D model to generate the first inverted 3D model
of the subsurface formation.
12. The system of claim 11, wherein the program code to apply the
constrained transformation further comprises program code
executable by the processor to cause the processor to apply a
constrained nonlinear transformation.
13. The system of claim 11, wherein the initial 3D model is a first
initial 3D model, and wherein the machine-readable medium further
comprises program code executable by the processor to cause the
processor to: generate a first lower-dimensional model, wherein the
first lower-dimensional model is at least one of a one-dimensional
(1D) model and a two-dimensional (2D) model of the subsurface
formation based on the value of the formation parameter, wherein
creating the first initial 3D model comprises using the first
lower-dimensional model; generate a second lower-dimensional model,
wherein the second lower-dimensional model is different from the
first lower-dimensional model, and wherein the second
lower-dimensional model is as at least one of the 1D model and the
2D model of the subsurface formation based on the value of the
formation parameter; generate a second initial 3D model based on
the second lower-dimensional model; generate a second transformed
3D model based the second initial 3D model using the unconstrained
minimization operation; and invert the second transformed 3D model
to generate a second inverted 3D model of the subsurface
formation.
14. The system of claim 11, wherein the machine-readable medium
further comprises program code executable by the processor to cause
the processor to generate a quality indicator corresponding with a
3D portion of the first inverted 3D model, wherein the quality
indicator is based on at least one of a standard deviation and
relative error of the 3D portion.
15. The system of claim 11, further comprising: a drill string in a
borehole; and a drill bit attached to the drill string, wherein the
machine-readable medium further comprises program code executable
by the processor to: determine a second formation parameter based
on the first inverted 3D model, and change a drilling direction or
drilling speed based on the second formation parameter.
16. One or more non-transitory machine-readable media comprising
program code to generate a first inverted 3D model of a subsurface
formation, the program code to: set a value of a formation
parameter for the subsurface formation; create an initial
three-dimensional (3D) model of the subsurface formation based on
the formation parameter; apply a constrained transformation to one
or more inversion variables of the initial 3D model to create a
variable-constrained 3D model of the subsurface formation; apply an
unconstrained minimization operation to the variable-constrained 3D
model to generate a first transformed 3D model; and invert the
first transformed 3D model to generate the first inverted 3D model
of the subsurface formation.
17. The one or more non-transitory machine-readable media of claim
16, wherein the program code to apply the constrained
transformation comprises program code to apply a constrained
nonlinear transformation.
18. The one or more non-transitory machine-readable media of claim
16, wherein the initial 3D model is a first initial 3D model, and
wherein the machine-readable media further comprises program code
to: generate a first lower-dimensional model, wherein the first
lower-dimensional model is at least one of a one-dimensional (1D)
model and a two-dimensional (2D) model of the subsurface formation
based on the value of the formation parameter, wherein creating the
first initial 3D model comprises using the first lower-dimensional
model; generate a second lower-dimensional model, wherein the
second lower-dimensional model is different from the first
lower-dimensional model, and wherein the second lower-dimensional
model is as at least one of the 1D model and the 2D model of the
subsurface formation based on the value of the formation parameter;
generate a second initial 3D model based on the second
lower-dimensional model; generate a second transformed 3D model
based the second initial 3D model using the unconstrained
minimization operation; and invert the second transformed 3D model
to generate a second inverted 3D model of the subsurface
formation.
19. The one or more non-transitory machine-readable media of claim
16, further comprising program code to generate a quality indicator
corresponding with a 3D portion of the first inverted 3D model,
wherein the quality indicator is based on at least one of a
standard deviation and relative error of the 3D portion.
20. The one or more non-transitory machine-readable media of claim
16, further comprising program code to drill a borehole into the
subsurface formation based on the first inverted 3D model.
Description
BACKGROUND
[0001] The disclosure generally relates to the field of underground
formation evaluation and, in particular, to using formation
resistivity measurements to determine petrophysical formation
properties.
[0002] Resistivity measurements and other sensor measurements are
used to determine the formation parameters of rock formations
surrounding a borehole. This information can be used to generate
three-dimensional (3D) models of the subsurface formation. More
accurate characterizations of a subsurface formation can include
formation parameters such as horizontal resistivity, vertical
resistivity, and anisotropy and increase the complexity of
generating an accurate 3D formation model. Such 3D models are
useful for a variety of purposes before, during, and after a well
operation.
BRIEF DESCRIPTION OF THE DRAWINGS
[0003] Embodiments of the disclosure may be better understood by
referencing the accompanying drawings.
[0004] FIG. 1 is a gridded three-dimensional (3D) model of a well
system.
[0005] FIG. 2 is a flowchart of operations to generate an inverted
3D model of a subsurface formation.
[0006] FIG. 3 is a cross-sectional view of a formation at a first
plane.
[0007] FIG. 4 is a cross-sectional view of a first 3D model
generated using a constrained transformation that shows the
resistivity of the formation at the first plane of the
formation.
[0008] FIG. 5 is a cross-sectional view of a second 3D model
generated using a 1D solver that shows the resistivity of the
formation at the first plane of the formation.
[0009] FIG. 6 is a cross-sectional view of a set of quality
indicators for a 3D model of the formation at the first plane
having no errors.
[0010] FIG. 7 is a cross-sectional view of a set of quality
indicators for a 3D model of the formation at the first plane
having some error.
[0011] FIG. 8 is a cross-sectional view of a formation at a second
plane.
[0012] FIG. 9 is a cross-sectional view of a third 3D model
generated using a constrained transformation that shows the
resistivity of the formation at the second plane of the
formation.
[0013] FIG. 10 is a cross-sectional view of a formation at a third
plane.
[0014] FIG. 11 is a cross-sectional view of a fourth 3D model
generated using a constrained transformation that shows the
resistivity of the formation at the first plane of the
formation.
[0015] FIG. 12 is an elevation view of an onshore platform
operating a downhole drilling assembly that includes a sensor to
measure values for a first formation parameter.
[0016] FIG. 13 is an elevation view of an onshore platform
operating a wireline tool that that includes a sensor to measure
values for a first formation parameter.
[0017] FIG. 14 depicts an example computer device.
DESCRIPTION OF EMBODIMENTS
[0018] The description that follows includes example systems,
methods, techniques, and operations that embody aspects of the
disclosure. However, it is understood that this disclosure may be
practiced without these specific details. For instance, this
disclosure refers to resistivity as a formation parameter. Aspects
of this disclosure can also be applied to other formation
parameters such as conductivity, density, heat conductivity,
permeability, porosity, etc. In other instances, well-known
structures and techniques have not been shown in detail in order
not to obfuscate the description.
[0019] Various embodiments include a set of operations and related
systems to generate a 3D model of subsurface formation and
determine formation parameters using the 3D model of the subsurface
formation. A system that includes a processor can generate a 3D
model of the subsurface formation by applying a nonlinear
minimization operation. In some embodiments, the system can
determine or otherwise set a value(s) for one or more initial
formation parameters, apply a nonlinear minimization operation to
generate a 3D model based on the initial formation parameters, and
invert the 3D model into an inverted 3D model useful for determine
resulting formation parameters. In some embodiments, the system can
use an unconstrained minimization operation, wherein an
unconstrained minimization operation does not apply an explicit
constraint on at least one formation parameter during the nonlinear
minimization operation.
[0020] After creating an initial 3D model of a subsurface formation
based on the value of a first formation parameter, the system can
apply a constrained transformation to one or more inversion
variables of the initial 3D model, wherein an inversion variable
can be a formation parameter or can be calculated from the first
formation parameter. For example, an inversion variable can be a
formation resistivity, anisotropy, formation dip, formation
azimuth, 3D model boundary, etc. The system can apply the
constrained transformations to one or more inversion variables of a
3D model to generate a variable-constrained 3D model of the
subsurface formation in order to constrain the set of possible
solutions when using an unconstrained minimization operation. The
system can then apply an unconstrained minimization operation to
the variable-constrained 3D model to generate a transformed 3D
model. The system can use the transformed 3D model by inverting the
model to generate an inverted 3D model of the subsurface
formation.
[0021] In some embodiments, the initial 3D model can be determined
based on a lower-dimensional model such as a one-dimensional (1D)
model or two-dimensional (2D) model. For example, the system can
first generate a 1D model of the subsurface based on an initial set
of resistivity measurements and use the 1D model to generate an
initial 3D model. In some embodiments, the system can generate more
than one lower-dimensional model. For example, the system can
generate a 1D model and a 2D model based on the value(s) of a
formation parameter and use the set of lower-dimensional models to
generate a set of initial 3D models. The system can then apply one
or more constrained transformations and an unconstrained
minimization operation to generate a first transformed 3D model and
a second transformed 3D model.
[0022] In some embodiments, the system can invert each of the
transformed 3D models and compare the results with each other to
generate a quality indicator correlated with the precision and/or
accuracy of the resulting formation parameters. In some
embodiments, a quality indicator can be assigned to each pixel of a
3D structure and determined based on standard deviation (SD),
relative error (RE), misfit log, 5% confidence, 50% confidence, 95%
confidence, etc. In some embodiments, the use of different
minimization operations can provide different 3D quality
indicators. For example, the system can use an unconstrained
minimization operation that includes the use of a Jacobian matrix
and a Gauss-Newton algorithm to produce a first quality indicator
corresponding with a first 3D model and use a 3D forward model to
produce second 3D quality indicator model. A 3D quality indicator
can provide users with an additional information useful for
performing well operations and/or drilling operations.
[0023] The operations described in this application can provide an
inverted 3D formation model having greater accuracy. In addition,
the ability to use an unconstrained minimization operation to
generate constrained solutions increases the robustness and
versatility of the operations for generating an inverted 3D
formation model to various data types across a variety of numeric
ranges. This inverted 3D formation can then be used for various
other operations, such as changing a drilling direction in response
to the detection of a target formation parameter determined using
the operations disclosed below.
Example 3D Model
[0024] FIG. 1 is a gridded three-dimensional (3D) model of a well
system The 3D model 100 of a well system can include multiple
pixels (sometimes also called "gridblocks," "elements," "cells,"
etc.), wherein each pixel represents a discrete space having the
same formation parameters within the discrete space. The 3D model
100 can be based on cartesian coordinates, wherein each pixel is
shown having a position along an x-axis 102, y-axis 104, and z-axis
106. In some embodiments, the pixels can have different sizes or
shapes from each other. In addition, the 3D model can be based on
different coordinate systems such as a cylindrical coordinate
system. The 3D model 100 can include a resistivity/conductivity
distribution. The resistivity distribution can be described as
being distributed in at least one of a zero-dimensional (0D), 1D,
2D, and/or 3D space. The distribution functions can be described in
various coordinate systems.
[0025] The 3D model 100 includes a first pixel 108 can correspond
with "limestone" as a material and have a corresponding first set
formation properties different from any of the other pixels. The 3D
model 100 can also include a second pixel 110 and third pixel 112,
wherein both pixels can correspond with a same material such as
"shale." In some embodiments, the second pixel 110 and third pixel
112 can share a set of formation properties. Alternatively, the
second pixel and third pixel 112 can have different formation
properties. For example, the second pixel 110 can have a
resistivity of 6.0 ohm-meter (ohm-m) while the third pixel can have
a resistivity of 0.8 ohm-m due to different water saturation values
between the two pixels. The 3D model 100 can also include the
fourth pixel 114 and fifth pixel 116, each of which can also have
their own formation parameters or share a subset of formation
parameters with another pixel. For example, the fourth pixel 114
and fifth pixel 116 can share a same vertical resistivity value but
have different horizontal resistivity values.
Example Flowchart
[0026] The flowchart below is provided to aid in understanding the
illustrations and are not to be used to limit scope of the claims.
The flowcharts depict example operations that can vary within the
scope of the claims. Additional operations may be performed; fewer
operations may be performed; the operations may be performed in
parallel; and the operations may be performed in a different order.
For example, the operations depicted in blocks 208-240 can be
performed in parallel or concurrently. It will be understood that
each block of the flowchart illustrations and/or block diagrams,
and combinations of blocks in the flowchart illustrations and/or
block diagrams, can be implemented by program code. The program
code may be provided to a processor of a general purpose computer,
special purpose computer, or other programmable machine or
apparatus.
[0027] FIG. 2 is a flowchart of operations to generate an inverted
3D model of a subsurface formation. The operations of the flowchart
200 are described with reference to a system that includes a
processor. The processor may execute the operations in the
flowchart 200 to cause the system to generate an inverted 3D model.
Operations of the flowchart 200 begin at block 204.
[0028] At block 204, the system sets the values of one or more
first formation parameters. The system can acquire values based
previously determined values from a reference table. Alternatively,
or in addition, the system can acquire measurements from one or
more of sensors in the wellbore of a formation or at the surface of
a formation. For example, with reference to FIG. 12 below,
measurements can be obtained from a sensor in the sensor tool 1210
and used to set the values of one or more first formation
parameters. These acquired values can then be used to set the one
or more first formation parameters such as resistivity.
[0029] At block 208, the system generates a set of initial 3D
models based on at least one 1D model and/or at least one 2D model.
The system can generate the 1D or 2D model based on a generalized
3D formation model, wherein the generalized 3D formation model can
be a formation conductivity model. The formation conductivity model
can be a 3D model having dimensions of
(N.sub.x.times.N.sub.y.times.N.sub.z), wherein N.sub.x is the total
number of pixels along the x-axis direction, N.sub.y is the total
number of pixels along the y-axis direction, and N.sub.z is the
total number of pixels along the z-axis direction. Every coordinate
point consisting of integers in the formation model can represents
a pixel. The conductivity of the formation model can be determined
using Equation 1 below, wherein .sigma..sub.ijk is the conductivity
of the formation represented by the coordinate at point i, j, and
k, and wherein and .PI..sub.ijk is a 3D boxcar function:
.sigma. .function. ( x , y , z ) .apprxeq. i = 1 N x .times. j = 1
N y .times. k = 1 N z .times. [ .sigma. i .times. j .times. k
.times. .times. .PI. i .times. j .times. k ] ( 1 ) ##EQU00001##
[0030] The 3D boxcar function can be determined based on three 1D
boxcar functions. In some embodiments, each of the boxcar boundary
limits can be infinite. For example, Equation 2 can have the
following boundary limits: xB.sub.1=-.infin.;
xB.sub.N.sub..chi..sub.+1=+.infin.; yB.sub.1=-.infin.,
yB.sub.N.sub.y.sub.+1=+.infin.; and zB.sub.1=-.infin.,
zB.sub.N.sub.z.sub.+1=+.infin., wherein .PI..sub.i(x, xB.sub.i,
xB.sub.i+1) is the 1D boxcar function along the x direction,
.PI..sub.j(y, yB.sub.j, yB.sub.j+1) is the 1D boxcar function along
the y direction, and .PI..sub.k(z, zB.sub.k, zB.sub.k+1) is the 1D
boxcar function along the z direction, wherein xB.sub.i represents
a boundary condition limit(s) in the x-direction for the pixels
wherein the x index value is equal to i, yB.sub.j represents a
boundary condition(s) in the y-direction for pixels wherein the y
index value is equal to j, and zB.sub.k represents a boundary
condition(s) in the z-direction for pixels wherein the z index
value is equal to k, respectively:
.PI..sub.ijk(x,y,z;xB.sub.i,xB.sub.i+1;yB.sub.j,yB.sub.j+1;zB.sub.k,zB.s-
ub.k+1)=.PI..sub.i(x,xB.sub.i,xB.sub.i+1).PI..sub.j(y,yB.sub.j,yB.sub.j+1)-
.PI..sub.k(z,zB.sub.k,zB.sub.k+1) (2)
[0031] In some embodiments, the formation can be anisotropic. For
example, the conductivities can be written as two 3.times.3
diagonal tensors for a transversely isotropic (TI) anisotropic
formation. The anisotropic quality of a formation can be modeled
with additional parameters such as an anisotropic ratio of the
formation conductivity for one or more of the pixels. The
anisotropic ratio of the formation conductivities can be determined
using two diagonal tensors, .sigma.(x, y, z) and .sigma..sub.ijk,
that describe the TI anisotropic formation, represented below in
Equations 3-5, wherein .sigma..sub.h is horizontal conductivity,
.sigma..sub.v is vertical conductivity, .sigma..sub.h.sup.ijk is a
horizontal conductivity of a pixel at (i, j, k), and
.sigma..sub.v.sup.ijk is a vertical conductivity of a pixel at (i,
j, k):
.sigma.(x,y,z)=diag(.sigma..sub.h(x,y,z),.sigma..sub.h(x,y,z),.sigma..su-
b.v(x,Y,z)) (3)
.sigma..sub.ijk=diag(.sigma..sub.h.sup.ijk,.sigma..sub.h.sup.ijk,.sigma.-
.sub.v.sup.ijk) (4)
[0032] In some embodiments, a conductivity anisotropic ratio
.sigma..sub.hv.sup.ijk of a pixel at (i, j, k) for its
corresponding formation conductivity can be defined by Equation 5
below:
.sigma. h .times. v i .times. j .times. k = .sigma. h i .times. j
.times. k .sigma. v i .times. j .times. k ( 5 ) ##EQU00002##
Furthermore, in some embodiments, a resistivity anisotropy can be
used to determine formation properties such as a formation dip and
azimuth angles (.alpha. & .beta.). Because conductivity is the
inverse of resistivity, the anisotropic ratio of the formation
resistivity of each pixel (i, j, k) can also be described as shown
in Equations 6 and 7, wherein R.sub.ijk is the resistivity
corresponding to the pixel at (i, j, k), R.sub.h.sup.ijk is the
horizontal resistivity component at (i,j,k), R.sub.v.sup.(ijk) is
the vertical resistivity component at (i,j,k), and
R.sub.vh.sup.(ijk) is the anisotropic ratio of resistivity of the
pixel at (i, j, k):
R.sub.ijk=diag(R.sub.h.sup.(ijk),R.sub.h.sup.(ijk),R.sub.v.sup.(ijk))
(6)
R v .times. h ( i .times. j .times. k ) = .sigma. h .times. v i
.times. j .times. k = .sigma. h i .times. j .times. k .sigma. v i
.times. j .times. k = R v ( i .times. j .times. k ) R h ( i .times.
j .times. k ) ( 7 ) ##EQU00003##
[0033] In some embodiments, the formation parameters can be
expressed as a column vector X (or 1-column matrix) as shown in
Equation 8 with total of N.sub.p3 parameters, wherein N.sub.p3 is
defined below for Equation 9, wherein R.sub.h.sup.m is the
horizontal resistivity component at the m-th pixel, R.sub.v.sup.m
is the vertical resistivity component at the m-th pixel, and
xB.sub.j is the boundary condition limit(s) in the x-direction for
the pixels wherein the x index value is equal to j:
X=(R.sub.h.sup.(1), . . .
,R.sub.h.sup.(N.sup.x.sup.N.sup.y.sup.N.sup.z.sup.), . . .
,R.sub.v.sup.(N.sup.x.sup.N.sup.y.sup.N.sup.z.sup.),xB.sub.2,xB.sub.N.sub-
..chi.,yB.sub.2,yB.sub.N.sub.y,zB.sub.2,zB.sub.N.sub.z,.alpha.,.beta.).sup-
.T (8)
N.sub.p3=[2N.sub..chi.N.sub.yN.sub.z+N.sub..chi.+N.sub.y+N.sub.z-1]
(9)
Equation 8 can be re-written by replacing the vertical
resistivities of pixels with their corresponding anisotropic ratios
to produce Equation 10 and its simplified form Equation 11, wherein
R.sub.h is the set of horizontal resistivity values
[R.sub.h.sup.(1), . . . ,
R.sub.h.sup.(N.sup..chi..sup.N.sup.y.sup.N.sup.z.sup.)] and
R.sub.vh is a set of resistivity anisotropy values
[R.sub.vh.sup.(1), . . . ,
R.sub.vh.sup.(N.sup..chi..sup.N.sup.y.sup.N.sup.z.sup.))], xB is a
set of the boundary condition limits [xB.sub.2, . . . ,
xB.sub.N.sub.X], yB is a set of the boundary condition limits
[yB.sub.2, . . . , yB.sub.N.sub.y], zB is a set of the boundary
condition limits [zB.sub.2, . . . , zB.sub.N.sub.z]:
X=(R.sub.h.sup.(1), . . .
,R.sub.h.sup.(N.sup..chi..sup.N.sup.y.sup.N.sup.z.sup.),R.sub.vh.sup.(1),
. . .
,R.sub.vh.sup.(N.sup..chi..sup.N.sup.y.sup.N.sup.z.sup.),xB.sub.2,
. . . ,xB.sub.2, . . . ,xB.sub.N.sub.x,yB.sub.2, . . .
,yB.sub.N.sub.y,zB.sub.2, . . .
,zB.sub.N.sub.z,.alpha..sub.t.beta.).sup.T (10)
X=(R.sub.h,R.sub.vh,xB,yB,zB,.alpha.,.beta.).sup.T (11)
[0034] The 3D pixel-based model, as represented by Equations 1-11
above, can be used to generate a 1D model approximation and/or a 2D
model approximation. In some embodiments, the 2D pixel-based model
can be generated by setting the constraints to the following:
N.sub.y=1, N.sub..chi.>1, N.sub.Z>1. For example, applying
these constraints to Equation 1 yields Equation 12, which
represents a 2D pixel-based model approximation for the
formation:
.sigma. .function. ( x , y , z ) .apprxeq. i = 1 N x .times. k = 1
N z .times. [ .sigma. i .times. j .times. k i .times. j .times. k ]
( 12 ) ##EQU00004##
[0035] The parameter vector can correspondingly be reduced to
Equations 13 and/or 14, wherein R.sub.h.sup.(1) is the horizontal
resistivity for the first pixel in the 2D model,
R.sub.h.sup.(N.sup..chi..sup.N.sup.z.sup.) is the horizontal
resistivity for the N.sub.xN.sub.z-th pixel in the 2D model,
R.sub.vh.sup.(1) is the resistivity anisotropy for the first pixel
in the 2D model, R.sub.vh.sup.(1) is the resistivity anisotropy for
the N.sub.xN.sub.Z-th pixel in the 2D model N.sub.x is the maximum
number of pixels in the x-direction, N.sub.Z is the maximum number
of pixels in the z-direction, xB.sub.N.sub.x is a boundary
condition limit in the x-direction for pixel(s) at the index value
x=N.sub.x, and zB.sub.N.sub.z is a boundary condition limit in the
z-th direction for pixel(s) at the index value z=Nz:
X=(R.sub.h.sup.(1),,R.sub.h.sup.(N.sup..chi..sup.N.sup.z.sup.),R.sub.vh.-
sup.(1),,R.sub.vh.sup.(N.sup..chi..sup.N.sup.z.sup.),xB.sub.2,xB.sub.N.sub-
.x,zB.sub.2,zB.sub.N.sub.Z,.alpha.,.beta.).sup.T (13)
X=(R.sub.h,R.sub.vh,xB,zB,.alpha..sub.t.beta.).sup.T (14)
[0036] Equations 12-14 can then be used to generate a 2D
conductivity approximation and its corresponding parameter vector.
The 2D pixel conductivity approximation, .sigma..sub.ik, can
expressed as function f.sub.ik.sup.(2D) of the boundary condition
limit yB.sub.j and the conductivity .sigma..sub.ijk as shown below
in Equation 15, wherein the 2D model approximation can have the
same parameters with the 3D model as shown above and N.sub.y is the
maximum number of pixels in the y-direction:
.sigma..sub.ik=f.sub.ik.sup.(2D)(yB.sub.j,.sigma..sub.ijk, j=1,2, .
. . ,N.sub.y), j=1,2, . . . ,N.sub.y (15)
[0037] In addition, the 3D pixel-based model as represented by
Equations 1-11 above can be used to generate a 1D model
approximation, which can sometimes be referred to as a 1D
multi-layer formation model. In some embodiments, the 1D
pixel-based model can be generated by setting the constraints to
the following: N.sub.y=1, N.sub.x=1, and N.sub.Z>1. For example,
applying these constraints to Equation 1 can yield Equation 16,
which represents a 1D pixel-based model approximation for the
formation conductivity .alpha.(z):
.sigma. .function. ( z ) .apprxeq. i = 1 N x .times. [ .sigma. k k
] ( 16 ) ##EQU00005##
[0038] The parameter vector X for a 1D model can correspondingly be
reduced to Equations 17 and/or 18, wherein R.sub.h.sup.(k) is the
horizontal resistivity for the k-th pixel in the 1D model,
R.sub.vh.sup.(k) is the resistivity anisotropy for the k-th pixel
in the 1D model:
X=(R.sub.h.sup.(1),,R.sub.h.sup.(k),,R.sub.h.sup.(N.sup.Z.sup.),R.sub.vh-
.sup.(1) . . .
,R.sub.vh.sup.(k),,R.sub.vh.sup.(N.sup.Z.sup.),zB.sub.2,,zB.sub.N.sub.Z,.-
alpha.,.beta.).sup.T (17)
X=(R.sub.h,R.sub.vh,zB, . . . ,.alpha.,.beta.).sup.T (18)
[0039] Equations 12-14 can then be used to generate a 1D
conductivity approximation and its corresponding parameter vector.
The 1D pixel conductivity approximation, .sigma..sub.k, can
expressed as function f.sub.k.sup.(1D) of the boundary condition
limits xB.sub.i and yB.sub.j and conductivity components
.sigma..sub.ijk as shown below in Equation 19:
.sigma..sub.k=f.sub.k.sup.(1D)(xB.sub.i,yB.sub.j,.sigma..sub.ijk,
i=1,2,,N.sub..chi., j=1,2,,N.sub.y) (19)
[0040] Using the 1D and 2D model represented by Equations 12-19,
the system can generate one or more initial 3D models to form a set
of initial 3D models. For example, the 1D model can be extended to
fill out the volume of an initial 3D model. In some embodiments,
the system can generate both a 1D model and a 2D model as
lower-dimension models, and then generate a first initial 3D model
based on the 1D model and generate a second initial 3D model based
on the 2D model. In addition, or alternatively, the system can use
other models such as a pre-run inversion model or an initial
setting from user-defined input data to generate one or more
additional initial 3D models. Alternatively, or in addition,
multiple initial guesses for multiple models that satisfy the
equality and inequality constraints on inversion variables
described below for blocks 212-220 can be randomly or
deterministically generated.
[0041] At block 212, the system applies a constrained
transformation to the pixel boundary values of each of the initial
3D models to generate a set of variable-constrained 3D models. In
some embodiments, the pixel boundary values can be transformed to
be constrained within a particular pixel boundary value range. For
example, each of the pixel boundary distances xb, yb, and zb can be
constrained to vary from -1 to 1. In addition, parameters of the
pixel boundary values can also be bound. For example, the bounds of
a measurement depth can be based on Equation 20, wherein B can be
any one of x.sub.B, y.sub.B, z.sub.B, and wherein B.sub.ref is the
corresponding coordinate at a reference point such as x.sub.Bref;
y.sub.Bref and Z.sub.Bref, and wherein DOI is a tool's depth of
investigation, wherein the tool was used to acquire measurement
values that the initial 3D model were based on:
|B-B.sub.ref|.ltoreq.DOI (20)
[0042] In some embodiments, the system can transform the pixel
boundary values using a constrained nonlinear transformation,
wherein the constrained nonlinear transform can be represented by
Equations 21 and 22 below, wherein X represents the set of
variables (x.sub.1, . . . , x.sub.N).sup.T and can be similar or
the identical to the variables described for Equation 11, and
wherein each variable x.sub.i can represent a formation parameter
such as resistivity, anisotropy, saturation, etc., and wherein X
represents a set of transformation variables (, . . . , ).sup.T,
wherein each of the values x, can be an inversion variable:
X=f({tilde over (X)} . . . ) (21)
{tilde over (X)}=(, . . . ,).sup.T (22)
[0043] In some embodiments, the constrained nonlinear
transformation f.sub.i can be represented by Equation 23 below,
wherein x.sub.l is one of the variables of X, and wherein
a.sub.i=x.sub.i.sup.min, and wherein
b.sub.i=(x.sub.i.sup.max-x.sub.i.sup.min), and wherein
x.sub.i.sup.min is the minimum value amongst the inversion
variables x.sub.i, and wherein x.sub.i.sup.max is the maximum value
amongst the inversion variables x.sub.i and
0.ltoreq.f.sub.i().ltoreq.1:
x.sub.i=f.sub.i({tilde over
(x)}.sub.l,a.sub.i,b.sub.i)=a.sub.i+b.sub.i*f.sub.i({tilde over
(x)}.sub.l) (23)
[0044] For example, the nonlinear transformation represented by
Equation 23 can be represented by (though not limited to) any one
of Equations 24-26, wherein x.sub.l is one of the variables of X,
and wherein k is a constant:
.times. f i .function. ( x l ) = 0.5 .function. [ sin .function. (
k .times. ) + 1 ] ( 24 ) .times. f i .function. ( x l ) = 1 + a k
.times. .times. ( 25 ) .times. f i .function. ( x l ) = sin 2
.function. ( k .times. ) ( 26 ) ##EQU00006##
[0045] In some embodiments, the system can transform the pixel
boundary values using a linear transformation. For a linear
transformation of a variable, the system can use Equations 27 and
28 below, wherein X is the variable before the linear
transformation, X.sub.tr is the variable after the linear
transformation, and A and B are two coefficients:
.times. X = X tr .times. A + B ( 27 ) .times. X tr = X - B A ( 28 )
##EQU00007##
[0046] The system can apply the transformations modeled using
Equations 27-28 above to determine an x-coordinate pixel boundary,
a y-coordinate pixel boundary, and a z-coordinate pixel boundary.
The boundaries can be represented by Equations 29-31 below, wherein
xb is a x-coordinate pixel boundary, yb is a y-coordinate pixel
boundary, zb a z-coordinate pixel boundary, x.sub.ref is an
x-coordinate reference point, y.sub.ref is ay-coordinate reference
point, z.sub.ref is a z-coordinate reference point:
xb = xB - x ref DOI ( 29 ) yb = yB - y ref DOI ( 30 ) zb = zB - z
ref DOI ( 31 ) ##EQU00008##
[0047] As will be described further below or block 230, the system
can use the boundaries above to invert very deep resistivity (VDR)
data from xb, yb, and zb to determine the boundary values in
physical units, wherein VDR data can include measurements of
formation parameters deeper than 5 meters into the formation from a
sensor. In some embodiments, the system can then recover the pixel
boundaries (xB, yB, zB) by using relationships based on
re-arrangements of Equations 29-31. For example, the system can
recover xB as shown in Equation 32 below, wherein xB a pixel
boundary in meaningful physical units:
xB=xb*DOI+x.sub.ref (32)
[0048] At block 216, the system applies a constrained
transformation to the resistivity values of each of the initial 3D
models to modify the set of variable-constrained 3D models. The
system can transform resistivity values by transforming horizontal
resistivity values, vertical resistivity values, and/or an
anisotropic ratio(s) of resistivity values. In some embodiments,
the resistivity values may be within one or more pre-determined
ranges. For example, the horizontal and vertical resistivity can be
within a range of 0.1 ohm-m to 1000 ohm-m. In some embodiments, the
ranges can be changed after the transformation. For example, using
the transformations represented by the equations below, vertical
resistivity value(s) R.sub.v can be constrained to a range between
-1 to 3 and the formation anisotropic ratio R.sub.vh can be
constrained to a range between 0 to 2. the system can determine
using Equation 33 below, wherein R.sub.h is the corresponding
inversion variable and is a constrained nonlinear transformation
function of R.sub.h, the minimum resistivity value Rhmin, and the
maximum resistivity value Rhmax:
=(R.sub.h,Rh min,Rh max) (33)
[0049] In the case where is a log transformation, Equation 33 can
be reduced to Equation 34 below:
= log a ( R h - Rhmin Rhmax - R h ) ( 34 ) ##EQU00009##
[0050] The VDR data can be used to determine and then one or more
horizontal resistivity values can be mapped back to formation
resistivity values in physical units as described below for block
236. For example, the system can invert to R.sub.h using the
equation 35 below, wherein a is the constant shown in Equation
34:
R h = Rhmin + ( Rhmax - Rhmin ) .times. ( 1 + ) ( 35 )
##EQU00010##
[0051] In some embodiments, can be a log transformation of only
R.sub.h, which results in Equations 36 and 37 below:
=log.sub.a(R.sub.h) (36)
R.sub.h= (37)
For example, R.sub.h can be equal to if the common logarithm is
used and R.sub.h can be equal to if the natural logarithm is
used.
[0052] The system can also apply a set of constrained
transformations to a first resistivity value to determine the
vertical resistivity value R.sub.v. In some embodiments, the system
can replace R.sub.v with the ratio R.sub.vh, wherein
R.sub.v/R.sub.h=R.sub.vh. R.sub.vh can be constrained to a
particular range. For example, R.sub.vh can be constrained to a
range of values greater than or equal to 1 and less than or equal
to 20. Alternatively, the formation anisotropic ratio R.sub.vh can
be constrained to a range of values greater than or equal to 0 and
less than or equal to 2. In some embodiments, the value for
R.sub.vh can be transformed into the constraint-transformed ratio
represented by Equation 38, wherein is a constrained nonlinear
transformation function of R.sub.vh, the minimum resistivity value
Rvhmin, and the maximum resistivity value Rvhmax:
=(R.sub.vh,Rvh min,Rvh max) (38)
[0053] In some embodiments, can include a log transformation of
R.sub.vh, Rvhmin, and Rvhmax, reducing Equation 38 to Equation 39
and its corresponding function to revert back by to R.sub.vh in
Equation 40 below:
= log a ( R vh - Rvhmin Rvhmax - R vh ) ( 39 ) R vh = Rvhmin + (
Rvhmax - Rvhmin ) .times. 2 + ( 40 ) ##EQU00011##
[0054] In some embodiments, can be a log transformation of
R.sub.vh, wherein the various relationships between , R.sub.vh, and
a can be represented below in Equations 41-43:
=log.sub.a(R.sub.vh) (41)
R.sub.vh= (42)
R.sub.v=*R.sub.h (43)
[0055] At block 220, the system applies a constrained
transformation to the dip and azimuth of each of the initial 3D
models to modify the set of variable-constrained 3D models. In some
embodiments, the system can use at least one of a linear and
nonlinear transformations to determine the dip and azimuth of a
formation or a portion of a formation. Alternatively, or in
addition, the system can use both a linear transformation and a
nonlinear transformation to determine the dip and azimuth. In some
embodiments, the dip and azimuth can be constrained to a range
between -90 degrees and +90 degrees, while the azimuth can be
constrained to be within a range between -180 degrees and +180
degrees. Alternatively, the dip and azimuth can be constrained to
be within a range between 0 degrees and +180 degrees, while the
azimuth can be constrained to be within a range between 0 degrees
and +360 degrees. In some embodiments, the ranges of the formation
dip and azimuth can be converted to radians. For example, the
formation dip can be constrained to a range between
- .pi. 2 .times. .times. and .times. + .pi. 2 . ##EQU00012##
The formation dip and azimuth can also be re-scaled and/or shifted.
For example, the formation dip can be transformed to range between
-1 and 1.
[0056] In some embodiments, the system can apply one or more
constrained nonlinear transformation such as those represented by
Equations 23-26 above. In some embodiments, the system can use a
relationship represented by Equations 44-45 to apply one or more
constrained linear transformation. For example, using Equation 28,
the system can set B equal to 0 and A equal to .alpha..sub.max to
result in a relationship represented by Equation 44 shown below.
Alternatively, or in addition, the system can use Equation 28 and
set B equal to 0 and A equal to .beta..sub.max to result in a
relationship represented by Equation 45 shown below. Equations 44
and 45 can then be used to apply a linear transformation for
determining a dip and azimuth:
.alpha. ' = .alpha. .alpha. max ( 44 ) .beta. ' = .beta. .beta. max
( 45 ) ##EQU00013##
[0057] Using Equations 44-45 above, if both dip and azimuth are in
degrees, the transformed dip .alpha.' and the transformed azimuth
.beta.' can be represented by Equations 46-47 below:
.alpha. ' = .alpha. .times. .pi. 180 ( 46 ) .beta. ' = .beta.
.times. .pi. 180 ( 47 ) ##EQU00014##
[0058] At block 224, the system generates a set of transformed 3D
models based on the variable-constrained 3D models by applying an
unconstrained minimization operation. In some embodiments, after
one or more inversion variables in a set of initial 3D model are
transformed to generate a set of variable-constrained 3D models,
the system can use nonlinear minimization operation to generate an
inverted 3D model. The nonlinear minimization operation can be used
to solve the problem represented by Equations 48-50 below, wherein
equation 48 can be reduced to Equation 50, O(X) is the cost
function of X=(x.sub.1,, x.sub.N).sup.T, X* is a solution of a cost
function minimization, and wherein the cost function and subsequent
solution(s) may be subject to equality constraints and/or
inequality constraints as shown in Equation 50, and wherein
X.sub.min and X.sub.max are the lower and upper bounds of X:
{ min .times. .times. O .function. ( X _ ) s . t . .times.
constraints .times. .times. ( e . g . , equality .times. .times.
and .times. .times. inequality .times. .times. constraints ( 48 ) X
* _ = ArgMin .times. { O .function. ( X _ ) ; s . t . .times. all
.times. .times. constraints } ( 49 ) { min .times. .times. O
.function. ( X _ ) s . t . .times. X _ min .ltoreq. X _ .ltoreq. X
_ max , xB i .ltoreq. xB i + 1 , yB j .ltoreq. yB j + 1 , zB k
.ltoreq. zB k + 1 ( 50 ) ##EQU00015##
[0059] The specific bounds pertinent to Equation 50 can be based on
the physical parameter bounds in the well logging domain for
hydrocarbon exploration. For example, as discussed for block 216,
the horizontal resistivity value R.sub.h and the vertical
resistivity value R.sub.v, respectively, can each be within a range
of being greater than or equal to 0.1 ohm-m and less than or equal
to 1000 ohm-m. In some embodiments, the nonlinear minimization
method can include a cost function calculation. The cost function
O(x) can be calculated using Equation 51, wherein the term .PHI.(X)
is a weighted residual square term and .phi..sub.reg(X) is a
regularization term:
O(X)=.PHI.(X)+.phi..sub.reg(X) (51)
[0060] In some embodiments, Equation 51 can be re-written as
Equation 52 below. In Equation 52, W.sub.d is a data weight
diagonal matrix and is equal to diag(w.sub.d1, . . . , w.sub.dj,,
w.sub.dM), w.sub.dj is an element of the data weight diagonal
matrix, W.sub.x is a model weight diagonal matrix and is equal to
diag(w.sub.x1, . . . , w.sub.xj, w.sub.xN), and X.sub.p is a
prescribed model parameter vector and X.sub.ave is defined by
Equation 53. In addition, .lamda.1 and .lamda.2 can be
regularization parameters that can serve as scalar values that
provide solution resolution, smoothing, function stability,
balancing minimization of data fitting error and regularization
terms. In some embodiments, .lamda.2 can be equal to zero to reduce
the cost function without the bound penalty term. In addition, the
normalized data misfit vector (X) can be defined using Equation 54,
wherein e.sub.j(X) is the misfit error for the j-th observed
data:
O .function. ( X _ ) = 1 2 .times. { W d _ _ e _ .function. ( X _ )
2 + [ .lamda. .times. .times. 1 W x _ _ ( X _ - X _ p ) 2 + .lamda.
.times. .times. 2 W x _ _ ( X _ - X _ ave ) 2 ] } ( 52 ) .times. X
_ ave = X _ min + X _ max 2 ( 53 ) .times. e _ .function. ( X _ ) =
( e 1 .function. ( X _ ) , , e j .function. ( X _ ) , , e M
.function. ( X _ ) ) 2 ( 54 ) ##EQU00016##
[0061] In some embodiments, the weighted residual square term
{.parallel.W.sub.d (X).parallel..sup.2 can be defined using
Equation 55, wherein d.sub.j.sup.obs is the j-th observed data, and
wherein d.sub.j.sup.pre is the j-th simulated data corresponding to
the j-th observed data, and wherein Nf.sup.(j) is a weighted
normalization function, and wherein x.sub.p is a prescribed value
for x:
W _ d _ e _ .function. ( X _ ) 2 = j = 1 M .times. w dj * e j
.function. ( X _ ) 2 = j = 1 M .times. w dj * d j obs - d j pre Nf
( j ) 2 ( 55 ) W _ _ x ( X _ - X _ p ) 2 = i = 1 N .times. w xi * (
X i - x pi ) 2 ( 56 ) W _ _ _ x ( X _ - X _ ave ) 2 = i = 1 N
.times. w xi * ( x i - x avei ) 2 ( 57 ) ##EQU00017##
[0062] In some embodiments, the system can use a Gauss-Newton
method to solve each of the plurality of 3D models. Applying the
Gauss-Newton method can include obtaining .DELTA.X.sub.n, which is
a step vector (i.e. direction vector) that corresponds with an
iteration n of an application of the Gauss-Newton method. The
system can use the step vector to update an inverted parameter
vector as shown in Equation 58 for cases wherein the
X.sub.min.ltoreq.X.sub.n+1.ltoreq.X.sub.max, wherein .gamma..sub.n
is a step length, X.sub.n is a parameter vector at time step n, and
X.sub.n+1 is an unknown parameter vector corresponding to time step
n+1:
X.sub.n+1=X.sub.n+.gamma..sub.n.DELTA.X.sub.n (58)
[0063] In some embodiments, the unknown parameter vector X.sub.n+1
can be updated using multiple nonlinear transformations. After
applying multiple nonlinear transformations, the transformation
result having the least misfit error can be selected as the
transformation result corresponding with that particular iteration
n. For example, multiple nonlinear transformations can be applied
using as shown in Equation 59, wherein f.sub.l(X.sub.n,
.gamma..sub.n, .DELTA.X.sub.n, X.sub.min, X.sub.max) is a nonlinear
function having an index value l, wherein X.sub.min and X.sub.max
are the lower and upper bounds of X, respectively:
X n + 1 l _ = f l .function. ( X _ n , .gamma. n , .DELTA. .times.
.times. X n _ , X _ min , X _ max ) , l = 1 , 2 , .times. , L ( 59
) ##EQU00018##
[0064] In some embodiments, the system can generate a sensitivity
matrix (i.e. Jacobian matrix) during a logging-while-drilling (LWD)
electromagnetic (EM) inversion. In some embodiments, the system can
determine the sensitivity matrix using at least one of a finite
difference (FD) method, an Adjoint method, a FD+adjoint method, and
a FD+adjoint+Broyden approximation. The system can use the
sensitivity matrix to increase the efficiency and accuracy of a
nonlinear minimization operation.
[0065] At block 230, the system generates a set of inverted 3D
models by inverting the transformed 3D models based on the
constrained transformations. The inverted 3D models can include
resulting formation parameters for one or more pixels in the 3D
model. For example, the inverted 3D model can include a resulting
formation parameter of 10 ohm-m for the resistivity of a pixel.
Inverting a transformed 3D model can be based on the type of
constrained transformation used. For example, to invert an
inversion variable to which a constrained nonlinear transformation
was applied, the system can use a nonlinear transformation. For
example, if the system used the nonlinear equation shown in
Equation 34 to apply a constrained transformation to the inversion
variable of an initial 3D model, the system can use the linear
equation shown in Equation 35 to invert the inversion variable when
generating the corresponding inverted 3D model. Alternatively, to
invert an inversion variable to which a constrained linear
transformation was applied, the system can apply a linear
transformation to invert the transformed 3D model. For example, if
the system used the linear equation shown in Equation 29 to apply a
constrained transformation to the inversion variable of an initial
3D model, the system can use the linear equation shown in Equation
32 to invert the inversion variable when generating the
corresponding inverted 3D model. In some embodiments, both linear
and nonlinear transformations can be applied to the inversion
variables of a transformed 3D model to generate an inverted 3D
model.
[0066] At block 236, the system can generate a set of quality
indicator values based on the inverted 3D models. In some
embodiments, applying the constrained transformation and nonlinear
minimization operations described above for a plurality of initial
3D models can result in multiple values for a same type of
formation parameter for a same portion of a 3D model, wherein the
portion can be a pixel, a set of pixels, or the entire 3D model.
For example, different nonlinear transformation results can provide
different resistivity values for the same pixel. In some
embodiments, the quality indicator can indicate an accuracy and/or
precision of a predicted formation parameter. In some embodiments,
the quality indicator can be determined from a standard deviation
(SD) and/or relative error (RE) calculation based on the multiple
values for a same type of formation parameter for some or all of
the pixels. Equation 60 represents a calculation the system can use
to determine a RE value, wherein .rho..sub.i is a parameter of
interest for an i-th element, .rho. is the average value for the
parameter of interest, and N is the total number of elements:
R .times. .times. E = 1 N .times. i = 1 N | .rho. i - .rho. .rho. |
, .rho. = 1 N .times. i = 1 N .times. .rho. i .times. .times. or
.times. .times. ( .rho. = ( i = 1 N .times. .rho. i ) 1 N ) ( 60 )
##EQU00019##
[0067] The system can provide the quality indicator values based on
the RE. For example, the system can use a relationship represented
by Equations 61 and 62, wherein Equation 61 corresponds with 1.0 as
representing no error and Equation 62 corresponds with 100 as
representing no error, and wherein w(Rt) is weighting factor and
QI2 represents a quality indicator value:
QI=w(Rt)[1.0-min(RE,1.0)] (61)
QI2=w(Rt)[100-min(RE,100)] (62)
[0068] In some embodiments, the quality indicator QI2 can be a
function of an intermediate quality indicator QI1, shown in
Equation 63, wherein misfit represents a misfit error. For example,
the relationship between QI2 and QI1 can be as shown in Equation
64:
.times. QI .times. .times. 1 = [ 1 - min .function. ( misfit , 1.0
) ] ( 63 ) QI .times. .times. 2 = QI .times. .times. 1 * [ 100 -
min .function. ( R .times. .times. E , 100 ) ] = [ 1 - min
.function. ( misfit , 1.0 ) ] * [ 100 - min .function. ( R .times.
.times. E , 100 ) ] ( 64 ) ##EQU00020##
[0069] At block 240, the system determines formation parameters
based on the inverted 3D models. In some embodiments, the result
provided by the inverted 3D models include a set of values that can
be used directly as the formation parameters. For example, the
average horizontal resistivity for a portion of a formation can be
calculated form the average of the horizontal resistivity values
corresponding to the pixel included in the portion of the
formation. Alternatively, or in addition, formation parameters can
be calculated from available formation parameters determined in the
3D model. For example, the system can implement a method following
Archie's equation (shown in Equation 65), wherein SW is a water
saturation, a and m are empirical constants, R.sub.w is a
resistivity of connate water in pore spaces of the formation,
R.sub.t is a formation resistivity:
S w n = aR w R t .times. .phi. m ( 65 ) ##EQU00021##
[0070] In some embodiments, Equation 64 can be modified by setting
the variables m and n to be equal to 2 and the variable a to be
equal to 1 in order to reduce Equation 64 to Equations 65 and 66.
Equation 65 can be used to determine a 3D porosity-saturation index
S.sub.w*.phi., and Equation 66 can be used to determine a 3D
saturation value SW (in the case of a known porosity value):
S w * .phi. = ( R w R t ) 1 2 ( 66 ) S w = 1 .phi. .times. ( R w R
t ) 1 2 ( 67 ) ##EQU00022##
[0071] At block 250, the system modifies a well operation based on
the resulting formation parameters. In some embodiments, the
resulting formation parameters are correlated with a formation
feature. For example, a set of resistivity values paired with a set
of density values can be correlated with hydrocarbon-rich region
the subsurface formation. In response, a drilling direction can be
changed to a new direction target the hydrocarbon-rich region and
drill a borehole in the new direction. As another example, a set of
resistivity values in the inverted 3D model can be correlated to a
water-rich region. In response, drilling direction can be changed
or drilling speed can be stopped/slowed to avoid the water-rich
region.
Example Results
[0072] FIG. 3 is a cross-sectional view of a formation at a first
plane The cross-sectional view of the formation 300 shows a top
formation layer 310, a middle formation layer 320 below the top
formation layer 310, and a lower formation layer 340 below the
middle formation layer 320. Each of the formation layers can have
different formation parameter values from each other. The top
formation layer 310 can be a homogeneous formation layer having a
resistivity of 5 ohm-m. The middle formation layer 320 can be an
inhomogeneous formation layer including three rock sections 321-323
and two water sections 331-332 in an alternating horizontal
arrangement along the middle formation layer 320. The lower
formation layer 340 can also have a resistivity of 5 ohm-m. In
addition, the formation 300 has a well path 306 that penetrates
each of the top formation layer 310, the middle formation layer
320, and the lower formation layer 340, wherein the well path 306
is shown at a 10 degree angle. While the well path 306 is shown at
a 10 degree angle in FIG. 3, the well path 306 can be in any other
direction.
[0073] Each of the layers can also include distinct sections having
their own resistivity values. For example, the three rock sections
321-323 have a resistivity of 50 ohm-m and the two water sections
331-332 have a resistivity of 0.5 ohm-m. In a real or simulated
environment, measurements from the sensors in the well path 306 can
provide sensor measurements such as EM measurements and acoustic
measurements. For example, a sensor at the sensor position 351 in
the well path 306 can be used to provide EM measurements.
[0074] FIG. 4 is a cross-sectional view of a first 3D model
generated using a constrained transformation that shows the
resistivity of the formation at the first plane of the formation.
FIG. 4 includes a 2D resistivity plot 400 that is based on an
inverted 3D model, wherein the inverted 3D model can be generated
using the operations described in the flowchart 200 of FIG. 2. The
formation resistivity shown in the plot region 410 can be similar
to the formation resistivity shown in the top formation layer 310
and the lower formation layer 340. The 2D resistivity plot 400
includes regions having various resistivity values and the same
well path 306 shown in FIG. 3. In addition, the 2D resistivity plot
400 includes a boxed region 405, which will be used for comparison
with the same boxed region 405 shown in a 2D resistivity plot 500
in FIG. 5 below.
[0075] The 2D resistivity plot 400 includes a set of
high-resistivity regions 421-423, wherein each of the
high-resistivity regions can be associated with a resistivity
greater than 15 ohm-m (e.g. 50 ohm-m). The 2D resistivity plot 400
also includes a low-resistivity region 431 at the same depth as and
between the high-resistivity regions 421 and 422, wherein the
low-resistivity region 431 has a resistivity less than 2 ohm-m
(e.g. 0.5 ohm-m). With reference to FIG. 3, the high-resistivity
region 421 can correspond with the rock section 321, the
high-resistivity region 422 can correspond with the rock section
322, and the high-resistivity region 423 can correspond with the
rock section 323. In addition, the low-resistivity region 431 can
correspond with the water section 331 and the low-resistivity
region 432 can correspond with the water section 332. In addition,
the angle of the well path 306 and sensors in the well path 306 do
not inhibit the operations described above from generating an
accurate inverted 3D model. For example, neither the
high-resistivity regions 421-423 or the low-resistivity regions
431-432 are at the same tilt as the well path 306.
[0076] FIG. 5 is a cross-sectional view of a second 3D model
generated using a 1D solver that shows the resistivity of the
formation at the first plane of the formation. A system can use a
1D solver such as a 1D forward modeling operation on the same
formation 300 shown in FIG. 3 to generate a 2D resistivity plot
500. The 2D resistivity plot 500 can be a 2D view of an inverted 3D
model using a 1D solver using measurements from the same well path
306. With reference to FIG. 4, a comparison between the
low-resistivity region 432 and the low-resistivity region 532 show
that predictions for formation parameters at regions close to the
well path 306 can be accurate. However, formation parameter
predictions for regions further away from the well path can be
significantly more accurate when using the operations similar to or
the same as those disclosed in the flowchart 200 of FIG. 2. For
example, with further reference to FIG. 4 whereas the plot
correctly shows the boxed region 405 having a resistivity less than
15 ohm-m (e.g. 10 ohm-m) in the 2D resistivity plot 400, the same
boxed region 405 can be shown having a resistivity greater than 15
ohm-m (e.g. 30 ohm-m) in the 2D resistivity plot 500. Moreover,
unlike the 3D model described in FIG. 4, the 1D solver generates
formation parameter predictions that shown to form a path that is
semi-parallel to the well path 306.
[0077] FIG. 6 is a cross-sectional view of a set of quality
indicators for a 3D model of the formation at the first plane
having no errors. As shown in the region 610, when the errors in an
area or volume of the 3D model are actually zero or artificially
set to zero, the quality indicator values corresponding to a
measurable area or volume to indicate no error. Furthermore, the
quality indicator values at a non-measured region 640 can be left
blank or otherwise indicate that no quality indicator value is
generated for the non-measured region 640. In some embodiments,
this can be indicated with a quantitative value such as 1.0 or
100%. In some embodiments, the quality indicator can indicate a
category or boolean value. For example, the quality indicator can
indicate that the system calculated the formation parameters
corresponding to a pixel with a confidence interval greater than
95%, 50%, or even 5%. Visually, the quality indicator can be
indicated using one or more colors, patterns, transparent overlays,
numbers, text, and/or some combination thereof.
[0078] FIG. 7 is a cross-sectional view of a set of quality
indicators for a 3D model of the formation at the first plane
having some error. The quality indicator plot 700 includes a first
region 710 and third region 730, wherein the quality indicator
values in the first region 710 and third region 730 show confidence
values greater than 90%. The quality indicator plot 700 also
includes a second region 720 between the first region 710 and the
third region 730, wherein the confidence values in the second
region 720 are less than 90%. The quality indicator plot 700 also
includes a fourth region 740 within the second region 720, wherein
the confidence values in the fourth region 740 are less than 80%.
In some embodiments, the quality indicator values can be based on a
relative error and/or standard deviation using operations similar
to or the same as those described for block 236 in FIG. 2
above.
[0079] FIG. 8 is a cross-sectional view of a formation at a second
plane. The cross-sectional view of a formation 800 shows a top
formation layer 810, a middle formation layer 820 below the top
formation layer 810, and a lower formation layer 840 below the
middle formation layer 820. Each of the formation layers can have
different formation parameter values from each other. The top
formation layer 810 can be a homogeneous formation layer having a
resistivity of 5 ohm-m. The middle formation layer 820 can be an
inhomogeneous formation layer including two rock sections 821-822
and a water section 831 in an alternating horizontal arrangement
along the middle formation layer 820. The lower formation layer 840
can also have a resistivity of 5 ohm-m. Each of the layers can also
include distinct portions having their own resistivity values. For
example, the two rock sections 821-822 can have a resistivity of 50
ohm-m and the water section 831 can have a resistivity of 0.5
ohm-m.
[0080] FIG. 9 is a cross-sectional view of a third 3D model
generated using a constrained transformation that shows the
resistivity of the formation at the second plane of the formation.
FIG. 9 includes a 2D resistivity plot 900 generated based on an
inverted 3D model, wherein the inverted 3D model can be generated
from the formation 800 using operations from the flowchart 200 in
FIG. 2. The formation resistivity shown in the plot region 910
accurately represents the formation resistivity shown in the top
formation layer 810 and the lower formation layer 840 as between 3
ohms-m and 15 ohms-m. Each of the high-resistivity regions 922-923
can be associated with a resistivity greater than 15 ohm-m (e.g. 50
ohm-m). With reference to FIG. 3, the high-resistivity region 922
can correspond with the rock section 822, the high-resistivity
region 923 can correspond with the rock section 823, and the water
sections 831-832 have a resistivity less than 2 ohm-m (e.g. 0.5
ohm-m).
[0081] FIG. 10 is a cross-sectional view of a formation at a third
plane. The cross-sectional view of a formation 1000 shows a top
formation layer 1010, a middle formation layer 1020 below the top
formation layer 1010, and a lower formation layer 1040 below the
middle formation layer 1020. Each of the formation layers can have
different formation parameter values from each other. The top
formation layer 1010 can be a homogeneous formation layer having a
resistivity of 5 ohm-m. The middle formation layer 1020 can be an
inhomogeneous formation layer including two rock sections 1021-1022
and a water section 1031 in an alternating horizontal arrangement
along the middle formation layer 1020. The lower formation layer
1040 can also have a resistivity of 5 ohm-m. Each of the layers can
also include distinct portions having their own resistivity values.
For example, the two rock sections 1021-1022 can have a resistivity
of 50 ohm-m and the water section 1031 can have a resistivity of
0.5 ohm-m.
[0082] FIG. 11 is a cross-sectional view of a fourth 3D model
generated using a constrained transformation that shows the
resistivity of the formation at the first plane of the formation.
FIG. 11 includes a 2D resistivity plot 1100 generated based on an
inverted 3D model, wherein the inverted 3D model can be generated
based on the formation 1000 of FIG. 10 using the flowchart 200 in
FIG. 2. The formation resistivity shown in the plot region 1110 can
be similar to the formation resistivity shown in the top formation
layer 1010 and the lower formation layer 1040. The 2D resistivity
plot 1100 includes a set of high-resistivity regions 1121-1122,
wherein each of the high-resistivity regions can be associated with
a resistivity greater than 15 ohm-m (e.g. 50 ohm-m). The 2D
resistivity plot 1100 also includes a low-resistivity region 1131
at the same depth as and between the high-resistivity regions 1121
and 1122, wherein the low resistivity section has a resistivity
less than 2 ohm-m (e.g. 0.5 ohm-m). With reference to FIG. 10, the
high-resistivity region 1121 can correspond with the rock section
1021, the high-resistivity region 1122 can correspond with the rock
section 1022, and the low-resistivity region 1131 can correspond
with the water section 1031.
Example Systems
[0083] FIG. 12 is an elevation view of an onshore platform
operating a downhole drilling assembly that includes a sensor to
measure values for a first formation parameter In FIG. 12, a
drilling system 1200 includes a drilling rig 1201 located at the
surface 1202 of a borehole 1203. The drilling system 1200 also
includes a pump 1250 that can be operated to pump mud through a
drill string 1204. The drill string 1204 can be operated for
drilling the borehole 1203 through the subsurface formation 1208
using the drill bit 1230.
[0084] The drilling system 1200 includes a sensor tool 1210 to
acquire sensor channel measurements from fluid and fluid mixtures
in the borehole, such as a pure formation fluid, a pure drilling
fluid, a mixture of formation fluid and drilling fluid, etc. The
sensor tool 1210 can be part of the drill string 1204 and lowered
into the borehole, optionally as part of a bottomhole assembly. The
sensor tool 1210 can include a set of EM sensors that can emit an
EM signal and/or receive an EM signal from the subsurface formation
1208. Alternatively, or in addition, the sensor tool 1210 can
include optical sensors, resistivity sensors, viscosity sensors,
density sensors, pressure sensors, etc. For example, the sensor
tool 1210 can include an optical sensor that detects pressure
measurements as the sensor tool 1210 is lowered into the
formation.
[0085] While or after the set of sensors acquire sensor
measurements, the computer 1255 can use the sensor measurements to
set the values of a set of first formation parameters based on the
sensor measurements. The computer 1255 can also generate an initial
3D model based on the set of first formation parameters. The
computer 1255 can also apply one or more constrained
transformations on the inversion variable(s) of the formation
parameters and generate a transformed 3D model. Moreover, in
response to determining that a portion of a formation includes a
target formation parameter corresponding with hydrocarbon presence
such as including an elevated resistivity correlated with low
density, drilling operations can be moved in the direction of the
formation having the target formation parameter. Alternatively, in
response to determining that a portion of formation corresponds
with the presence of water, such as by determining that a portion
of the formation has a low resistivity, drilling operations and/or
the drill bit 1230 can be altered or stopped.
[0086] FIG. 13 is an elevation view of an onshore platform
operating a wireline tool that that includes a sensor to measure
values for a first formation parameter. The onshore platform 1300
comprises a drilling platform 1304 installed over a borehole 1312.
The drilling platform 1304 is equipped with a derrick 1306 that
supports a hoist 1308. The hoist 1308 supports the wireline tool
1310 via the conveyance 1314, wherein specific embodiments of the
conveyance 1314 can be slickline, coiled tubing, piping, downhole
tractor, or a combination thereof. The wireline tool 1310 can be
lowered by the conveyance 1314 into the borehole 1312. Typically,
the wireline tool 1310 is lowered to the bottom of the region of
interest and subsequently pulled upward at a substantially constant
speed.
[0087] The wireline tool 1310 is suspended in the borehole by a
conveyance 1314 that connects the wireline tool 1310 to a surface
system 1318 (which can also include a display 1320). In some
embodiments, with reference to FIG. 12 above, the wireline tool
1310 can include one or more sensors analogous to the sensors
described as included in the sensor tool 1210. The wireline tool
1310 can include a set of EM sensors that can emit an EM signal
and/or receive an EM signal from the subsurface formation 1328.
Alternatively, or in addition, the wireline tool 1310 can include
optical sensors, resistivity sensors, viscosity sensors, density
sensors, pressure sensors, etc. For example, the wireline tool 1310
can include an optical sensor that detects density measurements as
the wireline tool 1310 is lowered into the subsurface formation
1328.
[0088] The sensor channel measurements can be communicated to a
surface system 1318 via the conveyance 1314 for storage,
processing, and analysis. The wireline tool 1310 can be deployed in
the borehole 1312 on coiled tubing, jointed drill pipe, hard-wired
drill pipe, or any other suitable deployment technique. In some
embodiments, the conveyance 1314 can include sensors to acquire
sensor measurements. The surface system 1318 can perform similarly
to the computer 1255 in FIG. 1 and generate an initial 3D model, a
variable-constrained 3D model, a transformed 3D model, etc. While
described as being performed by the computer 1255 or the surface
system 1318 at the surface, some or all of these operations can be
performed downhole and/or at a location that is remote to a well
location.
Example Computer
[0089] FIG. 14 depicts an example computer device A computer device
1400 includes a processor 1401 (possibly including multiple
processors, multiple cores, multiple nodes, and/or implementing
multi-threading, etc.). The computer device 1400 includes a memory
1407. The memory 1407 can be system memory. For example, the memory
1407 can include one or more of cache, SRAM, DRAM, zero capacitor
RAM, Twin Transistor RAM, eDRAM, EDO RAM, DDR RAM, EEPROM, NRAM,
RRAM, SONOS, PRAM, etc. or any one or more of the above already
described possible realizations of machine-readable media. The
computer device 1400 also includes a bus 1403. For example, the bus
1403 can include a PCI, ISA, PCI-Express, HyperTransport.RTM. bus,
InfiniBand.RTM. bus, NuBus, etc. The system can also include a
network interface 1405. For example, the network interface 1405 can
include a Fiber Channel interface, an Ethernet interface, an
internet small computer system interface, SONET interface, wireless
interface, etc.
[0090] The computer device 1400 includes a 3D model generator 1411
and an operations controller 1413. The 3D model generator 1411 can
perform one or more operations described above. For example, the 3D
model generator 1411 can apply a constrained transformation on one
or more inversion variables of a 3D model and generate an inverted
3D model. The operations controller 1413 can control or otherwise
modify drilling parameters or downhole operational parameters. For
example, the operations controller 1413 can instruct a drill bit to
stop rotating or change directions.
[0091] Any one of the previously described functionalities can be
partially (or entirely) implemented in hardware and/or on the
processor 1401. For example, the functionality can be implemented
with an application specific integrated circuit, in logic
implemented in the processor 1401, in a co-processor on a
peripheral device or card, etc. Further, realizations can include
fewer or additional components not illustrated in FIG. 14. For
example, the computer device 1400 can include one or more video
cards, audio cards, additional network interfaces, peripheral
devices, etc. The processor 1401 and the network interface 1405 are
coupled to the bus 1403. Although illustrated as being coupled to
the bus 1403, the memory 1407 can be coupled to the processor 1401.
The computer device 1400 can be a device at the surface and/or
integrated into component(s) in the borehole.
[0092] As will be appreciated, aspects of the disclosure can be
embodied as a system, method or program code/instructions stored in
one or more machine-readable media. Aspects can take the form of
hardware, software (including firmware, resident software,
micro-code, etc.), or a combination of software and hardware
aspects that can all generally be referred to herein as a "circuit"
or "system." The functionality presented as individual units in the
example illustrations can be organized differently in accordance
with any one of platform (operating system and/or hardware),
application ecosystem, interfaces, programmer preferences,
programming language, administrator preferences, etc.
[0093] Any combination of one or more machine readable medium(s)
can be utilized. The machine-readable medium can be a
machine-readable signal medium or a machine-readable storage
medium. A machine-readable storage medium can be, for example, but
not limited to, a system, apparatus, or device, that employs any
one of or combination of electronic, magnetic, optical,
electromagnetic, infrared, or semiconductor technology to store
program code. More specific examples (a non-exhaustive list) of the
machine-readable storage medium would include the following: a
portable computer diskette, a hard disk, a random access memory
(RAM), a read-only memory (ROM), an erasable programmable read-only
memory (EPROM or Flash memory), a portable compact disc read-only
memory (CD-ROM), an optical storage device, a magnetic storage
device, or any suitable combination of the foregoing. In the
context of this document, a machine-readable storage medium can be
any tangible medium that can contain or store a program for use by
or in connection with an instruction execution system, apparatus,
or device. A machine-readable storage medium is not a
machine-readable signal medium.
[0094] A machine-readable signal medium can include a propagated
data signal with machine readable program code embodied therein,
for example, in baseband or as part of a carrier wave. Such a
propagated signal can take any of a variety of forms, including,
but not limited to, electro-magnetic, optical, or any suitable
combination thereof. A machine-readable signal medium can be any
machine readable medium that is not a machine-readable storage
medium and that can communicate, propagate, or transport a program
for use by or in connection with an instruction execution system,
apparatus, or device.
[0095] Program code embodied on a machine-readable medium can be
transmitted using any appropriate medium, including but not limited
to wireless, wireline, optical fiber cable, RF, etc., or any
suitable combination of the foregoing.
[0096] Computer program code for carrying out operations for
aspects of the disclosure can be written in any combination of one
or more programming languages, including an object oriented
programming language such as the Java.RTM. programming language,
C++ or the like; a dynamic programming language such as Python; a
scripting language such as Perl programming language or PowerShell
script language; and conventional procedural programming languages,
such as the "C" programming language or similar programming
languages. The program code can execute entirely on a stand-alone
machine, can execute in a distributed manner across multiple
machines, and can execute on one machine while providing results
and or accepting input on another machine.
Additional Terminology and Variations
[0097] The program code/instructions can also be stored in a
machine-readable medium that can direct a machine to function in a
particular manner, such that the instructions stored in the
machine-readable medium produce an article of manufacture including
instructions which implement the function/act specified in the
flowchart and/or block diagram block or blocks.
[0098] Plural instances may be provided for components, operations
or structures described herein as a single instance. Finally,
boundaries between various components, operations and data stores
are somewhat arbitrary, and particular operations are illustrated
in the context of specific illustrative configurations. Other
allocations of functionality are envisioned and may fall within the
scope of the disclosure. In general, structures and functionality
presented as separate components in the example configurations may
be implemented as a combined structure or component. Similarly,
structures and functionality presented as a single component may be
implemented as separate components. These and other variations,
modifications, additions, and improvements may fall within the
scope of the disclosure.
[0099] Use of the phrase "at least one of" preceding a list with
the conjunction "and" should not be treated as an exclusive list
and should not be construed as a list of categories with one item
from each category, unless specifically stated otherwise. A clause
that recites "at least one of A, B, and C" can be infringed with
only one of the listed items, multiple of the listed items, and one
or more of the items in the list and another item not listed. A set
of items can have only one item or more than one item. As used
herein, the term "or" is inclusive unless otherwise explicitly
noted. Thus, the phrase "at least one of A, B, or C" is satisfied
by any element from the set {A, B, C} or any combination thereof,
including multiples of any element. Use of the phrase "a set of"
followed by an element should not be treated as exclusive to a
plurality of elements, unless specifically stated otherwise. A
clause that recites "a set of items" can be referring to one item
or a plurality of items.
EXAMPLE EMBODIMENTS
[0100] Example embodiments include the following:
[0101] Embodiment 1: A method comprising: setting a value of a
formation parameter for a subsurface formation; creating an initial
three-dimensional (3D) model of the subsurface formation based on
the formation parameter; applying a constrained transformation to
one or more inversion variables of the initial 3D model to create a
variable-constrained 3D model of the subsurface formation; applying
an unconstrained minimization operation to the variable-constrained
3D model to generate a first transformed 3D model; and inverting
the first transformed 3D model to generate a first inverted 3D
model of the subsurface formation.
[0102] Embodiment 2: The method of Embodiment 1, wherein applying
the constrained transformation comprises applying a constrained
nonlinear transformation.
[0103] Embodiment 3: The method of Embodiments 1 or 2, wherein the
one or more inversion variables comprise at least one of a
horizontal resistivity, a vertical resistivity, and an anisotropy
of the subsurface formation.
[0104] Embodiment 4: The method of any of Embodiments 1-3, wherein
the one or more inversion variables comprise at least one of a
formation dip and an azimuth of the subsurface formation.
[0105] Embodiment 5: The method of any of Embodiments 1-4, wherein
the one or more inversion variables comprise a boundary of a
portion of the first inverted 3D model.
[0106] Embodiment 6: The method of any of Embodiments 1-5, further
comprising drilling a borehole into the subsurface formation based
on the first inverted 3D model.
[0107] Embodiment 7: The method of any of Embodiments 1-6, wherein
the initial 3D model is a first initial 3D model, and wherein the
method of claim 1 further comprises: generating a first
lower-dimensional model, wherein the first lower-dimensional model
is at least one of a one-dimensional (1D) model and a
two-dimensional (2D) model of the subsurface formation based on the
value of the formation parameter, wherein creating the first
initial 3D model comprises using the first lower-dimensional model;
generating a second lower-dimensional model, wherein the second
lower-dimensional model is different from the first
lower-dimensional model, and wherein the second lower-dimensional
model is as at least one of the 1D model and the 2D model of the
subsurface formation based on the value of the formation parameter;
generating a second initial 3D model based on the second
lower-dimensional model; generating a second transformed 3D model
based the second initial 3D model using the unconstrained
minimization operation; and inverting the second transformed 3D
model to generate a second inverted 3D model of the subsurface
formation.
[0108] Embodiment 8: The method of any of Embodiments 1-7, further
comprises generating a quality indicator corresponding with a 3D
portion of the first inverted 3D model, wherein the quality
indicator is based on at least one of a standard deviation and
relative error of the 3D portion.
[0109] Embodiment 9: The method of any of Embodiments 1-8, wherein
the unconstrained minimization operation comprises using a
Gauss-Newton algorithm.
[0110] Embodiment 10: The method of any of Embodiments 1-9, wherein
the unconstrained minimization operation comprises using a Jacobian
matrix, wherein the Jacobian matrix is generated using at least one
of a finite difference method, an Adjoint method, a Broyden
approximation, and a combination thereof.
[0111] Embodiment 11: A system to generate a first inverted 3D
model of a subsurface formation, the system comprising: a
processor; and a machine-readable medium having program code
executable by the processor to cause the processor to, set a value
of a formation parameter for the subsurface formation; create an
initial three-dimensional (3D) model of the subsurface formation
based on the formation parameter; apply a constrained
transformation to one or more inversion variables of the initial 3D
model to create a variable-constrained 3D model of the subsurface
formation; apply an unconstrained minimization operation to the
variable-constrained 3D model to generate a first transformed 3D
model; and invert the first transformed 3D model to generate the
first inverted 3D model of the subsurface formation.
[0112] Embodiment 12: The system of Embodiment 11, wherein the
program code to apply the constrained transformation further
comprises program code executable by the processor to cause the
processor to apply a constrained nonlinear transformation.
[0113] Embodiment 13: The system of Embodiments 11 or 12, wherein
the initial 3D model is a first initial 3D model, and wherein the
machine-readable medium further comprises program code executable
by the processor to cause the processor to: generate a first
lower-dimensional model, wherein the first lower-dimensional model
is at least one of a one-dimensional (1D) model and a
two-dimensional (2D) model of the subsurface formation based on the
value of the formation parameter, wherein creating the first
initial 3D model comprises using the first lower-dimensional model;
generate a second lower-dimensional model, wherein the second
lower-dimensional model is different from the first
lower-dimensional model, and wherein the second lower-dimensional
model is as at least one of the 1D model and the 2D model of the
subsurface formation based on the value of the formation parameter;
generate a second initial 3D model based on the second
lower-dimensional model; generate a second transformed 3D model
based the second initial 3D model using the unconstrained
minimization operation; and invert the second transformed 3D model
to generate a second inverted 3D model of the subsurface
formation.
[0114] Embodiment 14: The system of any of Embodiments 11-13,
wherein the machine-readable medium further comprises program code
executable by the processor to cause the processor to generate a
quality indicator corresponding with a 3D portion of the first
inverted 3D model, wherein the quality indicator is based on at
least one of a standard deviation and relative error of the 3D
portion.
[0115] Embodiment 15: The system of any of Embodiments 11-14,
further comprising: a drill string in a borehole; and a drill bit
attached to the drill string, wherein the machine-readable medium
further comprises program code executable by the processor to:
determine a second formation parameter based on the first inverted
3D model, and change a drilling direction or drilling speed based
on the second formation parameter.
[0116] Embodiment 16: One or more non-transitory machine-readable
media comprising program code to generate a first inverted 3D model
of a subsurface formation, the program code to: set a value of a
formation parameter for the subsurface formation; create an initial
three-dimensional (3D) model of the subsurface formation based on
the formation parameter; apply a constrained transformation to one
or more inversion variables of the initial 3D model to create a
variable-constrained 3D model of the subsurface formation; apply an
unconstrained minimization operation to the variable-constrained 3D
model to generate a first transformed 3D model; and invert the
first transformed 3D model to generate the first inverted 3D model
of the subsurface formation.
[0117] Embodiment 17: The one or more non-transitory
machine-readable media of Embodiment 16, wherein the program code
to apply the constrained transformation comprises program code to
apply a constrained nonlinear transformation.
[0118] Embodiment 18: The one or more non-transitory
machine-readable media of Embodiments 16 or 17, wherein the initial
3D model is a first initial 3D model, and wherein the
machine-readable media further comprises program code to: generate
a first lower-dimensional model, wherein the first
lower-dimensional model is at least one of a one-dimensional (1D)
model and a two-dimensional (2D) model of the subsurface formation
based on the value of the formation parameter, wherein creating the
first initial 3D model comprises using the first lower-dimensional
model; generate a second lower-dimensional model, wherein the
second lower-dimensional model is different from the first
lower-dimensional model, and wherein the second lower-dimensional
model is as at least one of the 1D model and the 2D model of the
subsurface formation based on the value of the formation parameter;
generate a second initial 3D model based on the second
lower-dimensional model; generate a second transformed 3D model
based the second initial 3D model using the unconstrained
minimization operation; and invert the second transformed 3D model
to generate a second inverted 3D model of the subsurface
formation.
[0119] Embodiment 19: The one or more non-transitory
machine-readable media of any of Embodiments 16-18, further
comprising program code to generate a quality indicator
corresponding with a 3D portion of the first inverted 3D model,
wherein the quality indicator is based on at least one of a
standard deviation and relative error of the 3D portion.
[0120] Embodiment 20: The one or more non-transitory
machine-readable media of any of Embodiments 16-19, further
comprising program code to drill a borehole into the subsurface
formation based on the first inverted 3D model.
* * * * *